I just finished reading The Goal by Eliyahu M. Goldratt. It's a good book, and from my understanding, pretty well known. Goldratt is a a physicist who turned his scientific scrutiny to production line management, and his book is about how mathematical thinking can help your business.
In it, he describes a game used to illustrate the flow of a production line. It goes like this:
There are five people sitting in a line with a few boxes of matches at one end. The goal is to move as many matches as possible to the end of the line. In one turn, the first first person in the line rolls a six sided die and moves that number of matches down the line. The next person then rolls the die and moves that number of matches down the line, and so on. That's one turn. You repeat that process a given number of times and then count the number of matches moved all the way through ("throughput") and count the matches that are still sitting in between the people waiting to be moved ("inventory"). You want to maximize throughput and minimize inventory build up.
To be explicit, if on the first turn the first player rolls a 4 and the second player rolls a 6, they can only move 4 because there's only 4 available to move. But if there are more matches waiting in the inventory (perhaps because the person before them is getting high rolls while they've been getting low rolls), then they take from that pile. So if there are 2 already in inventory, player 1 rolls a 4 and player 2 rolls a 6, there are 6 available matches, so they can move all 6 down the line.
Hopefully that's clear. Here is the question, or rather questions:
- What is the expected value for throughput after $n$ turns?
- What about the expected inventory waiting for each person after $n$ turns?
- How does the number of people affect these values?
- What if the dice are more abstract, like a 17 sided die with four 6s, three 12s, etc. Just an abstract probability distribution?
- What about an expected value for the amount of rolls that were "wasted"? So if there was 2 in inventory and you rolled a 5, that would be 3 units "wasted"
- Can we find more details about the distribution, rather than just the expected value?